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Properties

Label 2-12e2-16.3-c4-0-2
Degree $2$
Conductor $144$
Sign $-0.819 + 0.573i$
Analytic cond. $14.8852$
Root an. cond. $3.85814$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.97 + 2.67i)2-s + (1.73 + 15.9i)4-s + (2.84 − 2.84i)5-s − 76.7·7-s + (−37.2 + 52.0i)8-s + (16.0 − 0.876i)10-s + (−121. − 121. i)11-s + (27.1 + 27.1i)13-s + (−228. − 205. i)14-s + (−249. + 55.2i)16-s + 88.0·17-s + (−261. + 261. i)19-s + (50.2 + 40.3i)20-s + (−37.3 − 686. i)22-s − 93.4·23-s + ⋯
L(s)  = 1  + (0.744 + 0.667i)2-s + (0.108 + 0.994i)4-s + (0.113 − 0.113i)5-s − 1.56·7-s + (−0.582 + 0.812i)8-s + (0.160 − 0.00876i)10-s + (−1.00 − 1.00i)11-s + (0.160 + 0.160i)13-s + (−1.16 − 1.04i)14-s + (−0.976 + 0.215i)16-s + 0.304·17-s + (−0.723 + 0.723i)19-s + (0.125 + 0.100i)20-s + (−0.0772 − 1.41i)22-s − 0.176·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(14.8852\)
Root analytic conductor: \(3.85814\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :2),\ -0.819 + 0.573i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.145005 - 0.460270i\)
\(L(\frac12)\) \(\approx\) \(0.145005 - 0.460270i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.97 - 2.67i)T \)
3 \( 1 \)
good5 \( 1 + (-2.84 + 2.84i)T - 625iT^{2} \)
7 \( 1 + 76.7T + 2.40e3T^{2} \)
11 \( 1 + (121. + 121. i)T + 1.46e4iT^{2} \)
13 \( 1 + (-27.1 - 27.1i)T + 2.85e4iT^{2} \)
17 \( 1 - 88.0T + 8.35e4T^{2} \)
19 \( 1 + (261. - 261. i)T - 1.30e5iT^{2} \)
23 \( 1 + 93.4T + 2.79e5T^{2} \)
29 \( 1 + (272. + 272. i)T + 7.07e5iT^{2} \)
31 \( 1 + 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + (1.04e3 - 1.04e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 915. iT - 2.82e6T^{2} \)
43 \( 1 + (-1.11e3 - 1.11e3i)T + 3.41e6iT^{2} \)
47 \( 1 - 1.72e3iT - 4.87e6T^{2} \)
53 \( 1 + (734. - 734. i)T - 7.89e6iT^{2} \)
59 \( 1 + (-1.20e3 - 1.20e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (-580. - 580. i)T + 1.38e7iT^{2} \)
67 \( 1 + (1.48e3 - 1.48e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 5.57e3T + 2.54e7T^{2} \)
73 \( 1 - 6.61e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.55e3 + 2.55e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.09e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.71e3T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.14950613001385045978985396561, −12.42804803281566492831575059824, −11.13073643993942495175486234591, −9.854745456147501762962729782564, −8.663411743176057724102866859506, −7.54093505206554800563375253844, −6.26683874012451269636621573973, −5.59859777713100856511214848741, −3.87902660999495751349244115950, −2.82227905672313522555408653171, 0.13712263965820740763824016922, 2.31560487697406706104674077743, 3.44954593440434304466378904757, 4.88850361491513857296205031500, 6.16560015959152423894303746982, 7.13086295684187932913852567654, 9.018429410326602439883325111636, 10.12012442187768343398443195878, 10.58498878183551913886042711297, 12.15818207622230604125471008666

Graph of the $Z$-function along the critical line